Ind. Eng. Chem. Res. 1991,30,2098-2104
2098
Analysis of Complex Reaction Networks in Gas-Liquid Systems: The Ozonation of 2-Hydroxypyridine in Aqueous Solutions Roberto Andreozzi,’ Vincenzo Caprio,*Maria G. D’Amore,*Amedeo Insola,+and Vincenzo Tufano*** Istituto di Ricerche sulla Combustione, Comiglio Nazionale delle Ricerche, and Dipartimento di Ingegneria Chimica, Uniuersitd di Napoli Federico 11, p.le V. Tecchio, 80125 Napoli, Italy
The ozonation of aqueous solutions of 2-hydroxypyridine is experimentally studied in a mechanically stirred batch reactor, and the reaction mechanism is assessed on the basis of chemical arguments. A mathematical model is developed that includes a complex reaction network and a description of the ozone transfer from the gas to the liquid in terms of pseudoreactions. The model is compared with the experimental data to assess its domain of validity, i.e., the range of initial concentration and stirrer speed in which the different causes of error are reduced to a negligible extent. In this range, the best estimate values of the unknown parameters (rate constants and coefficient of mass transfer) are computed. The behavior of the mass-transfer coefficient is discussed as a function of the initial concentration and of the stirrer speed, thus giving some insight on the interaction between mass transfer and chemical reaction in gas-liquid systems.
Introduction In the study of complex reactive systems, the exhaustive treatment of the experimental informations by means of adequate mathematical models and of rigorous statistical criteria can allow not only correct evaluation of the kinetic parameters, which represents the basis for a safe scale-up of the laboratory data to the industrial scale, but also achievement of a better understanding of the reaction mechanisms through the rational comparison among alternative reaction schemes. These arguments make the modeling procedure particularly attractive in the case of gas-liquid systems in which a complex reaction network is coupled with the mass transfer of a reactant from gas to liquid. This coupling often determines the failure of more simplified methods of modeling and of data analysis. This paper discusses a methodology that can be used when the reaction mechanism is safely singled out on the basis of sound chemical arguments. The ozonation of 2-hyroxypyridine in aqueous solution is chosen as a test system because this requirement is fulfilled, as reported elsewhere (D’Amore, 1990). Experimental Apparatus and Procedure Aqueous solutions of 2-hydroxypyridine (HPYR) of different initial concentration C, (up to 5.2 mM) buffered at pH = 5 by means of KH2POl and Na2HP04are submitted to ozonation at 20 OC in a 1090-cm3glass batch reactor. In order to prevent any radicalic development of the oxidation process tert-butyl alcohol (20 mM) is added to the reacting solutions as a radical scavenger. Mixing is obtained by means of a blade stirrer operated at different velocities (380 < n < 600 rpm). An ozonized oxygen stream (about 4% by volume) is fed through a sintered glass disk at a flow rate of 600 cm3/min into 800 cm3 of reacting solution. A more detailed description of the experimental apparatus, which is schematically shown in Figure 1, has been reported elsewhere (D’Amore, 1990). The instantaneous absorption rate of ozone is evaluated by continuous UV (A = 253 nm) monitoring of its concentration in the outlet gaseous stream. The chemical evolution of the ozonization process is followed by means Coneiglio Nazionale delle Ricerche. di Napoli Federico 11.
4 Universiti
of chemical and instrumental (HPLC, UV) analyses of the reacting solution at varying ozonation time. Care is taken to quench the reaction by rapidly cooling and degassing the analytical samples. The HPYR concentration is determined by means of W measurements (A = 325 nm), whereas the intermediates and products are quantitatively evaluated by HPLC analyses. The samples of the reacting mixture and some isolated products are also submitted to iodometric tests to quantitatively reveal the presence of active oxygen. The identification of compounds is generally performed by comparing their retention times and W spectra with those of pure substances. The identification of reaction intermediates for which no chemical standard is available is performed according to the following procedures. 1Formyl-5-hydroxy-3-pyrrolin-2-one is identified by means of NMR and MS analyses of the isolated compound, acid is recwhereas 4-(N-formyl)amino-4-oxo-2-butenoic ognized through the identification of its fragmentation products deriving from acid-catalyzed hydrolysis. Analytical Procedures are also adopted in order to obtain indications on the liquid-phase concentration of ozone and hydrogen peroxide.
Mathematical Model The mathematical model deals with a complex reaction network (chemical submodel) coupled with Ozone transfer from the bubbles to the liquid phase (fluid dymmic submodel). Two main hypotheses are assumed, namely, (a) the liquid phase is perfectly mixed and (b) the kinetic regime of absorption with reaction holds. Operating conditions consistent with the first assumption are derived from a previous work (D’Amore and Tufano, 1991)in which the experimental reactor was modeled as a set of three well-mixed reactors (liquid phase, L, bubbles, B, gaseous freeboard, F). According to the hypothesis of kinetic regime of absorption with reaction, the reaction is assumed to occur entirely in the well-mixed liquid bulk. The chemical reaction in the liquid film, although assumed negligible in the equations of mass balance, produces an increase of the mass-transfer coefficient (kLa> kLoa;Astarita et al., 1983). Since it is not possible to quantify a priori this effect, it is accounted for by considering kLa as an adjustable parameter, kept constant for each run.
0888-5885/91/2630-2098$02.50/0 Q 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30, No. 9, 1991 2099 Table I. Kinetic Equations of the Chemical Submodel R, = k,(HPYR)(Om 1 Ra kJHYDP) .. R; = k;(HYDPj R; = L;(HPYR)(O&) R3 k&FPJW(O&L) Re = ke(OarS(O8L) RI = ~,(FOBA)(OSL) RB = ke(H202)(08~) & = kdFPYR)(OaL) RIO= kio(HzOz)
-_
4
A and Q) must be singled out by comparing several chemically sound alternatives. Therefore, the model must be used in the identification mode, to compute the best estimates of the unknown kinetic parameters through a nonlinear regression technique. The multiple weighed least-squares function
+p u.v.
Nln Ne
4=
Figure 1. Experimentalapparatus (schematic). 1, oxygen cylinder; 2, flowmeter; 3, ozonizer; 4, UV spectrophotometer;5, ozone decomposition unit; 6, stirred vessel; 7, thermostat; 8, thermometer; 9, stirrer; 10, optoelectronicprobe; 11, gas sparger; A, B, valves.
On the basis of the above discussed hypotheses, the equations of mass balance are written as N,
dY,/dt = C[A,$,] =F, ,=I
for c = l-N,
(1)
where the stoichiometric coefficients A,, are assumed negative for the reactants and positive for the products. The absolute reaction rate, R,, is computed as Nc
R, = k,C[Y,"..]
for r = 1 4 ,
c=l
(2)
where the vector k of the rate constants is univocally defined by the matrix A. The N, equations (1)and (2) represent the chemical submodel, which must be coupled with the fluid dynamic submodel, written in terms of the following pseudoreactions: = Nr + 1 k, = k ~ a k(3)~ (VL/VB)& 0 3 ~
- -
0 3 ~ '
(VL/VB)O~B ?' = Nr
(VF/VB)OQB O3F
+ 2 k, = k
~ d
(4)
= Nr
+3
k, = l/tF
(5)
r = N,
+4
k,
(6)
0 3 ~ P
l/tF
which involve ozone in the liquid phase (03L) and two additional components, namely, ozone in the bubbles (Om) and in the freeboard (Os).Pseudoreaction 6 accounts for the unreacted ozone leaving the reador. At the same time, the term QCo/VB,which represents the ozone entering the reactor, must be added to the mass balance on 03B.The factors (VL/VB) and (VF/VB) are introduced as pseudostoichiometric coefficients to account for the different volumes of the model reactors used to define the concentrations YB and Yp, whereas the equilibrium constant kH is taken from Caprio et al. (1982). In conclusion, the mathematical model takes the form of a set of ordinary differential equations whose initial conditions (7) Y, = COc for t = 0 and c = l-N, + 2 are determined by the experimental values of concentration at the initial time. The integration of the model allows simulation of the behavior of the reacting system, for any assigned reaction network, which is defined by the values of A, k,and Q. In general, not only are most of the rate constants not known a priori, but the same reaction scheme (i.e., the matrices
I3
C[Yme- CmeI'Wme
m = l e=1
(8)
is minimized with respect to the parameters (8)by the Marquardt technique (Reklaitis et al., 1983). Since the functions Y,(t) are not available in analytical form, the required partial sensitivities for m = l-N, and p = l-Np (9) S, = aY,/d@, are computed through the N a p equations: = dF,/d@p = dS,/dt N,
aFm/Wp + C [dFm/aycI[ayc/WpI (10) ,=I
for which the initial conditions at t = 0 S, = 0 for c = l-N, and p = l-Np
(11)
hold because the initial values of concentration do not depend on the values of the parameters. Complex reaction networks are usually solved by means of shortcut procedures, the most widely used being the HJB method (Himmelbau et al., 1967). Nevertheless, the more rigorous procedure previously discussed appeared more appropriate because it has been demonstrated that the use of simplified methods may result in noticeable errors in the computed best-estimate values of the parameters (Hosten, 1979). It must be underlined that the proposed procedure easily allows inclusion of components in the reaction network for which no experimental measurement of concentration is available or, with better reason, components for which the experimental measurements are lacking at some experimental time. This is the case of ozone in the liquid phase and in the bubbles, and of hydrogen peroxide. Chemical Characterization of the System Because of the presence of a radical scavenger (tert-butyl alcohol), the reaction uniquely proceeds via the simpler direct mechanism. On the basis of the chemical arguments discussed hereafter, the reaction network shown in Figure 2 and in Table I was developed. The following discussion is also based on the observation of the experimental C(t) patterns. Results from a typical ozonation experiment are shown in Figure 6 under Results and Discussion. The electrophilic attack of ozone to the aromatic ring of the reactant (HPYR) determines the formation of two reactive unsaturated intermediates of both cyclic and open structure, namely, l-formyl-5-hydroxy-3-pyrrolin-2-one (FPYR) and 4-(N-formyl)amino-4-oxo-2-butenoic acid (FOBA), with remarkably higher production of the first one. Hydrogen peroxide is also formed as a side product of the first reaction. The following oxidation of both of these two intermediates produces N-formyloxamic acid (FOXA), which in
2100 Ind. Eng. Chem. Res., Vol. 30, No. 9, 1991
OAH
O A H FOBA
HYDP
I
I
.
H
[SI
03 0 3 Ha02 +
H202
0
20
30 t, mil
O‘H
FOXA
[91
10
Figure 3. Ozone concentration Cp measured in the stream leaving the reactor as a function of time for n = 500 rpm and Werent initial concentrations. (a) Co = 2.03 mM. (b)Co = 2.81 mM. (c) Co = 3.80 mM. (d) Co = 5.15 mM.
~
D 01
Figure 2. Reaction network. HPYR = 2-hydroxypyridine; FPYR = l-formyl-5-hydroxy-3-pyrrolin-2-one; PERX = peroxidic product, FOBA = 4-(N-formyl)amino-4-oxo-2-butenoic acid; HDYP = hydroperoxidic intermediate; FOXA = N-formyloxamic acid.
the experimental conditions adopted is stable to further attack by ozone, and two peroxidic species. The components were separated by HPLC and submitted to ozonation. The results indicate that the peroxidic species only derive from FPYR, whereas FOXA is produced from both direct oxidation of FOBA and from the spontaneous decomposition of one of the peroxidic intermediates. This intermediate (HDYP) is likely to possess a hydroperoxidic structure, since its decomposition also produces hydrogen peroxide. The second peroxidic species (PERX) exhibits a good stability under conditions of both spontaneous and ozone-activated decomposition, whereas the slow HDYP decomposition also contributes to ita formation, aa shown by the measured concentration profiles. Thus, the active oxygen content remains almost constant in the final ozonation stages. Production of ammonia through the acid-catalyzed hydrolysis of both HDYP and PERX indicates that these species have an amidic structure. Proposed structures of the peroxidic intermediates are shown in Figure 2. The cyclic structure of PERX and its high stability are also supported by literature indications concerning the production of similar species (Perry, 1959). The quantitative evaluation of both HDYP and PERX, performed on the basis of the active oxygen content of the reacting mixture, closely agrees, as expected, with the overall nitrogen balance. Therefore, the occurrence of any other significant reaction should be excluded. Nevertheless, the experimental observations suggest the inclusion in the reaction network of additional reactions that involve only ozone and hydrogen peroxide. In all the experiments the ozone concentration in the stream leaving the reactor, CF,moves at high reaction times to asymptotic values, CF-,which are lower than the inlet concentration ,C thus suggesting the importance of the ozone self-decomposition reaction. A second-order reaction has been included (reaction 8, Figure 2) according to the literature evidence in the case of moderately acidic solutions (Hewes and Davison, 1971; Sotelo et al., 1987) and
o.*v -
0 0
10
20
30 t,min
Figure 4. Ozone concentration Cp measured in the stream leaving the reactor as a function of time for Co = 2.8 mM and different stirrer speeds. (a) n = 380 rpm. (b)n = 450 rpm. (c) n = 600 rpm.
in the presence of a radical scavenger (Nadezhdin, 1988). The experimental C F ( t )patterns are reported for different initial concentrations Coand stirrer speeds n, respectively in Figures 3 and 4. The plotted data show that CF.. decreases on increasing Co,while being almost independent of n. This behavior suggests that the ozone decomposition is, at least partially, promoted by some of the species present in the reacting system. Hydrogen peroxide, which is formed by reaction 1in Figure 2, can exert this role since it is known to act as a catalyst at the lowest concentrations (Bray, 1938) and as a reactant at the highest concentrations (Staehelin and HoignB, 1982). This justifies the inclusion of reaction 9 (Figure 2). Different oxidable species, such as aldehydic intermediates, are present in the reacting system. Therefore, an additional consumption of hydrogen peroxide must be considered, which is accounted for by the overall pseudofirst-order reaction 10 (Figure 2).
Results and Discussion The experimental data were analyzed by the mathematical model to assess the domain of validity, i.e., the range of initial concentration and stirrer speed in which the different causes of error are reduced to a negligible extent, and to compute the beat estimate values of the rate constants. In the second part of this section, some results are also discussed on the effect of the chemical reaction on the ozone mass-transfer coefficient. Assessment of the Domain of Validity and Evaluation of the Rate Constants. The domain of validity is
Ind. Eng. Chem. Res., Vol. 30,No.9, 1991 2101 Table 11. Statistical Indices, Percent Saturation, SlIr,and Best Estimate Values of the Parameters, Computed at Constant Initial Concentration (C, = 2.8 mM) and Different Stirrer Speeds n run no. 6O 7b 7 8 9 n, rpm 380 450 450 500 600
R2,% 103uy,mM 103uF,mM s1/29
kLa, min-l kl'
k,e kaC klC k,' 103k,d 103k,d hae k,C 103k a
95.53 89.0 13.0 10.6 8.95 f 0.50 37.3 f 9.3 5.1 f 1.7 7.0 f 1.4 7.9 f 3.2 3.91 f 0.83 9.1 f 6.2 6.3 f 7.8 2.94 f 0.27 1.85 f 0.36 93 f 17
99.89 13.7 10.3 16.0 13.33 f 0.41 35.1 f 2.4 4.03 f 0.32 6.82 f 0.41 6.91 f 0.67 4.05 f 0.25 8.0 f 1.3 10.2 f 1.6 2.34 f 0.13 1.11 f 0.13 105 i 11
99.91 12.0 14.6 18.9 14.79 f 0.49 31.8 f 1.9 3.64 f 0.24 6.18 f 0.34 6.63 f 0.51 3.69 f 0.20 8.00 f 0.85 10.4 f 1.1 2.29 f 0.17 1.10 f 0.16 100 f 14
99.8 14.8 4.7 19.8 18.10 f 0.90 36.6 f 3.0 3.54 f 0.33 7.87 f 0.61 8.76 f 0.99 4.52 f 0.36 12.7 f 1.1 5.6 f 1.3 2.38 f 0.26 1.36 f 0.20 81 f 13
99.90 13.0 11.9 20.1 19.50 f 0.83 39.5 f 2.6 4.93 f 0.36 7.44 f 0.46 7.82 f 0.65 4.36 f 0.27 6.88 f 0.89 10.5 f 1.1 2.27 f 0.23 1.24 f 0.19 97 f 15
W,= 36. * W,= 4. 'm3/(mol min). dmin-l.
Table 111. Statistical Indices, Percent Saturation Sllr,and Best Estimate Values of the Parameters, Computed at Constant Stirrer Speed (n = 500 rpm) and Different Initial Concentrations Co run no. 5 8 11 12 CO,mM 2.03 2.80 3.80 5.15
RZ,% 103uy,mM 103uF,mM S1/2r
kLa, m i d k2O
kt ksU 1oSkeb 1 0 3 ~ kea kVU 103k12
99.77 15.1 13.3 22.6 16.13 f 0.94 38.1 f 3.3 5.03 f 0.50 7.39 f 0.57 8.02 f 0.91 4.42 f 0.35 9.7 f 1.9 12.6 f 2.3 0.84 f 0.52 0.82 f 0.20 58 f 36
99.86 14.8 4.7 19.8 18.10 f 0.90 36.6 f 3.0 3.54 f 0.33 7.87 f 0.61 8.76 f 0.99 4.52 f 0.36 12.7 f 1.1 5.6 f 1.3 2.38 f 0.26 1.36 f 0.20 81 f 13
99.64 32.0 22.5 17.4 20.5 f 1.5 37.2 f 5.1 3.44 f 0.53 7.41 f 0.97 8.2 f 1.6 4.00 f 0.54 12.2 f 1.9 0.7 f 2.6 2.6 f 1.9 0.58 f 0.30 38 f 51
99.53 51.1 33.6 12.8 24.2 f 2.6 45 f 11 4.9 f 1.2 9.0 f 2.1 9.0 f 2.4 5.3 f 1.2 8.1 f 1.6 10.8 f 2.0 3.4 f 1.1 1.36 f 0.70 68 f 25
"m3/(molmin). bmin-l.
assessed as a function of n and Cotwhose effects are respectively shown in Tables I1 and 111. This domain is shown in Figure 5 together with the experimental plan performed. At the highest values of n and a t constant Co (n 1 450 rpm, Table 11) a fairly good fitting is obtained, as shown by the values of the correlation coefficient R2and of the overall standard deviation by,,which amounts to less than 3% of the average concentration. A minimum is observed for the standard deviation on ozone, uF, at n = 500 rpm. In these conditions, the value assumed by the ratio R, = uY/uF(Ro N 3) agrees with the higher accuracy expected for the measurements of 03F. Lower values of R,, such as those obtained at n = 450 and at n = 600 rpm, indicate a less satisfactory correlation. In effect, uF is the most sensible index of the inadequacies of the model, Le., of the presence of some systematic mismatching between the experimental and the computed concentrations attributable to the partial failure of one or more hypotheses of the model. This is because the YF(t) curve is computed on the basis of the total consumption rate of ozone, which is only partially determined by the reactions that concern the substrate, whose rate constants (kl-k6)may be evaluated with a high accuracy on the basis of a plentiful and accurate set of experimental data. A nonnegligible fraction of the ozone is consumed by the reactions of decomposition, whose rate constants (ks-klo) are not supported by direct experimental data since the measurements of ozone in the liquid (YL)and of
mM -
Figure 5. Experimental plan and domain of validity aa a function of stirrer speed n and of initial concentration C,.
hydrogen peroxide are lacking. Therefore, the above defined inadequacies particularly affect the values of YL(t) and, consequently, of YF(t)rather than the concentration profiles of the other measured components. In this range of n, the residuals do not show any systematic error and are evenly distributed among the different components, as can be argued from Figure 6, in which the computed Y ( t )curves are compared with the experimental data, for run 8. The profiles computed for YLand H202 do not disagree with the available (qualitative) experimental informations, which were not included
2102 Ind. Eng. Chem. Res., Vol. 30, No. 9,1991 Table IV. Statistical Indices, Percent Saturation SI and Best Estimate Values of the Parameters, Computed at Low Stirrer Sbeeds n and at Low Initial Concentration
d
1 0.85 380 99.80 6.9 6.8 15.8 5.22 f 0.15 45.8 f 3.4 5.98 f 0.53 8.77 f 0.51 8.11 f 0.81 4.75 f 0.29 7.7 f 2.4 15.7 f 2.7 0.437 f 0.083 1.34 f 0.13 61 f 15
run no. CA.mM n, rpm
R2.%
loby,mM l@aF,mM
SlI2t
%
kLa, min-' k1" k2'
k 3" k4' 6"
lek2
103k,* k a" k9"
103klob
2 1.21 300 99.57 13.1 14.0 9.5 3.334 f 0.090 37.2 f 4.2 4.49 0.59 6.87 f 0.69 9.2 f 1.5 3.80 f 0.40 7.9 f 2.5 10.6 i 3.1 0.24 f 0.52 0.77 f 0.25 41 f 51
4 1.98 380 99.90 9.0 8.1 11.8 6.55 f 0.12 34.9 f 2.0 3.95 f 0.26 6.96 f 0.35 6.59 f 0.50 3.81 f 0.20 8.79 f 0.89 8.2 f 1.1 1.04 f 0.14 1.05 f 0.13 81 f 12
3 1.22 380 99.80 9.5 10.6 14.8 5.53 f 0.15 36.7 f 2.7 4.52 f 0.40 6.94 f 0.44 7.57 f 0.81 3.82 f 0.25 10.1 f 1.7 10.1 f 2.0 0.49 f 0.13 1.17 f 0.16 77 h 19
*
"m3/(mo1 min). bmin-'.
Table V. Overall Best Estimate Values of the Rate Constants kl" 36.7 f 1.5 lek2 9.4 f 2.1 k2' k3'
k,'
kb'
1.51
i c h
7 i
'll&l
0.5
0
10
20 t , m i n
Figure 6. Run 8: measured (C) and computed (Y) concentrations as a function of time t . (A) Left scale: & ozone leaving the reactor; dashed line, ozone entering the reactor. Right scale: ozone in the liquid phase. (B) Left scale: O, 4-(N-formyl)-amin~~oxo-2-butenoic acid (FOBA); B, N-formyloxamic acid (FOXA). Right scale: hydrogen peroxide. (C)Left scale: n,S-hydroxypyridine (HPYR); A, l-fomyl-5-hydroxypyolin-2-one (FPYR);V, hydroperoxidic intermediate (HDYP); B, peroxidic product (PERX).
in the experimental set. Similar results have been obtained for the other runs. The values of kLa and of the percent saturation defined as the ratio between YL and the corresponding value at the gas-liquid interface, computed at the time in which the HPYR concentration equals C0/2,show the expected asymptotic behavior as a function of n. The rate constants are evaluated with different accuracies, as measured by the relevant standard deviations, which also give a first indication of the sensitivity of the model to the single parameters. The numerical program can be forced to comply with the expected condition R , > 1 by assuming W , > 1 (weighed identification). For example, by setting WF= 4, the results relative to run 7 are modified as shown in the third column of Table 11. At the lowest values of n examined, a fair estimate of the parameters and an acceptable value of fsF (but less satisfactory values of R2and of by) can be only obtained by the weighed identification and rather high values of W , (run 6). Both the statistical indices and the rate constants present a residual sensitivity on WFthat makes less reliable the results obtained with this procedure. Therefore, the value n = 450 rpm marks a bound of the domain of validity of the model, which appears to be rather sharp possibly because in this range the mass-transfer coefficient strongly
4.20 f 0.60 7.17 f 0.37 7.63 f 0.91 4.08 f 0.30
k9'
lekl*
8.6 f 3.8 1.05 f 0.27 83 27
"m3/(moi min). *mi&.
depends on n. The occurrence of this bound may be tentatively explained by the establishment of the diffusive regime of reaction in the first stages of reaction because of the low values of kLa. The effect of Co is reported in Table 111. At a stirrer speed located imide the domain of validity ( n = 500 rpm), the overall statistical indices, R2 and cry, remain almost constant at the lowest values of Co,whereas the correlation becomes poorer on increasing C,. The errors observed at the highest values of Comay be explained by the nonnegligible chemical reaction in the liquid film, whose occurrence is clearly demonstrated by the increase of kLu observed on increasing C, The reaction in the film occurs at values of YLhigher than the bulk values, thus resulting in an apparent increase of the rate constants. The bound of the domain of validity is in this case smoother, poseibly because the consumption rate of ozone is a linear function of c,. The domain of validity extends at lower values of n and Co, as demonstrated by the results reported in Table IV. The slightly higher values of some rate constants observed at the lowest Coexamined (run 1) may be tentatively attributed to the difficulty of effectively quenching the reaction in the samples, which is very fast because of the high ozone concentration. The above discussed results suggest that the reported runs (with the exception of runs 6,10, and 12) are located inside the domain of validity of the model, as previously defined. For each rate constant, the relevant ouerall best estimate was computed as a weighed average of the valuea taken from Tables II-IV, the weighta being inversely proportional to the single standard deviations, and the values were reported in Table V. This procedure was not applied to k8 because ita dependence on C, is confirmed by the experimental data reported in Figures 3 and 4. The values obtained do not disagree with the literature data (Gurol and Singer, 1982; Sotelo et al., 1987), which are rather scattered because the ozone self-decomposition appears to be very sensitive to
Ind. Eng. Chem. Res., Vol. 30, No. 9, 1991 2103
,I
a
'/-;J, ,
,
0
2
4
0
Co.mM
Figure 7. Ozone mass transfer coefficient kLa as a function of the initial concentration Co for different stirrer speeds n. A, n = 300 rpm; 0, n = 380 rpm; 0, n = 450 rpm; 0,n = 500 rpm; V, n = 600
".
different variables, including water impurities. The value obtained for the only rate constant directly comparable with literature (k,) agrees fairly well with the available data (Staehelin and HoignB, 1982). No literature data are available for the other rate constants. The variability of the parameters around the overall best estimates (Table V) is very similar to the accuracy bk of the single estimates (Tables 11-IV), the most accurate being the estimates of kl,kS,kq,and kg. Different reasons must be invoked to explain the lower accuracy observed for the other rate constants. Reaction 2 (Figure 2) occurs in parallel with reaction 1 and only accounts for a small fraction of the total consumption rate of the HPYR. Similarly, the contribution of reactions 6 and 7 in the equations of mass balance is significant only at the highest reaction times. Therefore, the constants k6 and k, are not completely independent (as demonstrated by the rather high values computed for the covariance) and their estimation procedure is rather sensitive to the experimental errors. In the case of kg and klo, the lower accuracy may be explained by recalling that their estimation procedure is not supported by direct experimental data. On the other hand, the inclusion of these reactions in the reaction network noticeably improves the estimates of the other rate constants (D'Amore, 1990). Effect of the Chemical Reaction on the Ozone Mass-Transfer Coefficient. The computed values of the mass-transfer coefficient give some insight on the interaction between mass transfer and chemical reaction. Figure 7 shows the behavior of kLa as a function of Co,for constant n. The value at n = 380 rpm and Co = 3.8 mM (run 10, not included in the tables) was obtained by means of a restricted identification. With this procedure, the rate constants kl k7 were fixed at the overall best estimate values and kLa together with the rate constants k8-klo were identified. The data obtained at n = 380 rpm suggest a very slight dependence of the mass-transfer coefficient on the initial concentration at the lowest values of Coand a stronger and almost linear dependence at the highest values. The data obtained at n = 500 rpm confirm the linear dependence and the relevant slope. Unfortunately, it was not possible to extend the experiments at low Co and high n because in these conditions the reaction is too fast to allow the chemical analyses. The extrapolation of these relationships at the limit Co 0 allows evaluation of two values of the physical mass-transfer coefficient kLoa and consequently the enhancement factor E = k L a / k ~ ' a attributable to a fast reaction effect. Obviously, this procedure gives a rather
-
u
0.1
0
200
400
n.rm
Figure 8. Measured and extrapolated ozone mass transfer coefficients kLoa and kLa as a function of the stirrer speed n. 0,bidistilled water; 0 , nonreacting solutions; reacting mixtures: e, values extrapolated at Co 0 from Figure 7; . , CO= 3.8 m M 0, Co= 2.8 mM, E, Co = 2.0 mM, O, Co = 1.2 mM.
-
reliable value at n = 380 rpm, whereas the value a t n = 500 rpm is affected by a higher uncertainty, which can be estimated to be about f2 mi&. In the range of validity, the values of E do not exceed 1.5. Thus, the constancy of k ~ during a each run appears to be a reliable hypothesis. In any case, only a more detailed analysis would allow assessment of the effect on the computed rate constants of the dependence of the masstransfer coefficient on the overall ozone reaction rate. The data presently obtained can be compared with the values of kLoawhich were independently measured. Figure 8 shows the values measured in bidistilled water (D'Amore and Tufano, 1991), in nonreactive solutions, obtained by the addition of the phosphoric salts and of the scavenger, and in reacting solutions. The data obtained in nonreactive solutions can be fairly well extended to include the values of kLoaextrapolated from Figure 7. At n = 500 rpm, the reported error bar accounts for the above discussed uncertainty. The values of kLoaand their dependence on the stirrer speed agrees fairly well with the literature data, when the effect of the ionic strength is accounted for (Van't Riet, 1979; Hassan and Robinson, 1980). According to the literature (Van't Riet, 1979), this effect increases on increasing n. The comparison between curve b and the values of kLa obtained for the reacting mixtures, which are plotted for different values of Co (curves c), strengthens the reliability of the function kLoa(n).
Conclusions The analysis of the experimental data by means of a detailed mathematical model of the reacting system allowed obtaining of significant resulta in the case examined, in which the reaction mechanism is supported by a plentiful and accurate set of experimental data and by sound chemical arguments. The rate constants were obtained, and the behavior of the mass-transfer coefficient was assessed as a function of the initial concentration and of the stirrer speed. These results represent a aound basis to attempt the analysis of reacting systems for which the chemistry involved is not completely understood. In this case, it is believed that mathematical modeling will contribute to the assessment of the best reaction mechanism, through comparison among different alternatives. Nomenclature a = effective mass transfer area per unit volume of the liquid phase, cm-'
2104 Ind. Eng. Chem. Res., Vol. 30, No. 9, 1991
A = stoichiometric coefficient C = measured concentration, mol cm-3 E = enhancement factor for mass transfer
F
= function k = rate constant kH = gas-liquid equilibrium constant, dimensionless kL = mass-transfer coefficient in reacting systems, cm min-' kLo = mass-transfer coefficient in nonreacting systems, cm min-' n = stirrer speed, rpm N , = number of components Ne = number of experimental times N,,, = number of measured components N = number of parameters I( = number of reactions Q = volumetric flow rate, cm3 min-' R = reaction rate, mol cm-3 min-' R2 = correlation coefficient S = partial sensitivity SlI2= percent saturation of the liquid phase t = time, min tp = mean residence time in the freeboard, min V = volume, cm3 Y = computed concentration, mol cm-3 W = weight Greek Symbols
8 = adjustable parameter
u = standard deviation 4 = least-squares objective function s2 = reaction order
Subscripts
B = bubbles c = component e = experimental F = gaseous freeboard L = liquid m = measured 0 = initial p = parameter r = reaction = asymptotic Registry No. 2-Hydroxypyridine, 142-08-5.
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Receiued for review December 19, 1990 Revised manuscript received April 25, 1991 Accepted May 20,1991
